where \(\hat{s} \)  labels the pointer states (see Refs. [1]}, [2]}
for discussion of pointer states). This form is a consequence of “branching”
[3]} and appears in the good decoherence
limit of purely decohering models, which will be extensively discussed
below. Here, it is sufficient to note that the state, Eq. (REF ),
is the most relevant to quantum Darwinism. It makes little difference
if one treats the \(\mathcal {A}\)  as \(\mathcal {E}/\mathcal {F}\)  or as just the fully decohered,
or directly measured, \(\mathcal {S}\) , even when \(\mathcal {F}\)  is extremely large in
absolute terms Only for “global” questions, where \(\mathcal {F}\)  is some sizable fraction
of the environment, does it matter. Since the environment is huge
for most problems of everyday interest, such as photon scattering,
\(\mathcal {F}\)  can be very large—even asymptotically large—without
concern for this. However, Eq. (REF ) does drop exponentially
small corrections in the size of \(\mathcal {E}/\mathcal {F}\)  and one can not formally
take the asymptotic limit of \(\mathcal {F}\)  without first doing so in \(\mathcal {E}\) .
The degree to which asymptotic approximations work thus relies on
the balance sheet—how well records are kept in the environment
components compared to \(\mathcal {E}\) 's absolute size. Ref. [4]}
has dealt with retaining corrections to Eq. (REF ).. Hereon, I will treat the auxiliary system \(\mathcal {A}\)  as if it were \(\mathcal {S}\) .
